Complex analysis is known as one of the classical branches of mathematics and analyses complex numbers concurrently with their functions, limits, derivatives, manipulation, and other mathematical properties. Complex analysis is a potent tool with an abruptly immense number of practical applications to solve physical problems. Let’s understand various components of complex analysis one by one here.

Complex numbers

A number of the form x + iy where x, y are real numbers and i² = -1 is called a complex number.

In other words, z = x + iy is the complex number such that the real part of z is x and is denoted by Re(z), whereas the imaginary part of z is iy and is denoted by I(z).

Modulus and Argument of a Complex Number

The modulus of a complex number z = x + iy is the real number √(x² + y²) and is denoted by |z|.

The amplitude or argument of a complex number z = x + iy is given by:

arg(z) = θ = tan^-1(y/x), where x, y ≠ 0.

Also, the arg(z) is called the principal argument when it satisfies the inequality -π < θ ≤ π, and it is denoted by Arg(z).

Click here to learn about the argument of complex numbers.

Complex Functions

In complex analysis, a complex function is a function defined from complex numbers to complex numbers. Alternatively, it is a function that includes a subset of the complex numbers as a domain and the complex numbers as a codomain. Mathematically, we can represent the definition of complex functions as given below:

A function f : C → C is called a complex function that can be written as

w = f(z), where z ∈ C and w ∈ Z.

Also, z = x + iy and w = u + iv such that u = u(x, y) and v = v(x, y). That means u and v are functions of x and y.

Limits of Complex Functions

Let w = f(z) be any function of z defined in a bounded closed domain D. Then the limit of f(z) as z approaches z₀ is denoted by “l”, and is written as

\begin{array}{l} lim_{z \to z_{0}} f (z) = l \end{array}

, i.e., for every ϵ > 0, there exists δ > 0 such that |f(z) – l| < ϵ whenever |z – z₀| < δ where ϵ and δ are arbitary small positive real numbers. Here, l is the simultaneous limit of f(z) as z → z₀.

Continuity of Complex Functions

Let’s understand what is the continuity of complex functions in complex analysis.

A complex function w = f(z) defined in the bounded closed domain D, is said to be continuous at a point Z0, if f(z0) is defined

\begin{array}{l} lim_{z \to z_{0}} f (z) \end{array}

exists and

\begin{array}{l} lim_{z \to z_{0}} f (z) = f (z_{0}) \end{array}

Complex Differentiation

Some of the standard results of complex differentiation are listed below:

dc/dz = 0; where c is a complex constant
d/dz (f ± g) = (df/dz) ± (dg/dz)
d/dz [c.f(z)] = c . (df/dz)
d/dz zⁿ = nz^n-1
d/dz (f.g) = f (dg/dz) + g (df/dz)
d/dz (f/g) = [g (df/dz) – f (dg/dz)]/ g²

All these formulas are used to solve various problems in complex analysis.

Analytic Functions

A function f(z) is said to be analytic at a point z₀ if f is differentiable not only at z, but also at every point in some neighbourhood of z₀. Analytic functions are also called regular, holomorphic, or monogenic functions.

Also, check:Analytic functions

Harmonic Function

A function u(x, y) is said to be a harmonic function if it satisfies the Laplace equation. Also, the real and imaginary parts of an analytic function are harmonic functions.

Complex Integration

Suppose f(z) be a function of complex variable defined in a domain D and “c” be the closed curve in the domain D.

Let f(z) = u(x, y) + i v(x, y)

Here, z = x + iy

(or)

f(z) = u + iv and dz = dx + i dy

∫_c f(z) dz = ∫_c (u + iv) dz

= ∫_c (u + iv) (dx + idy)

= ∫_c (udx – vdy) +i ∫_c (udy + vdx)

Here, ∫_c f(z) dz is known as the contour integral.

Cauchy’s Integral Theorem

If f(z) is analytic function in a simply-connected region R, then ∫_c f(z) dz = 0 for every closed contour c contained in R.

Converse:

If a function f(z) is continuous throughout the simple connected domain D and if ∫_c f(z) dz = 0 for every closed contour c in D, then f(z) is analytic in D.

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